Hello, today, we're going to talk about
The Radiative Transfer Equation and Absorption and Emission Spectroscopy. In spectroscopy, we typically use focused
beams of electromagnetic radiation looking for how the radiation is absorbed,
emitted, or scattered. The radiation may come from a laser or
a conventional light source. There's a number of possibilities. We track the propagation of radiation
using the radiative transfer equation. And that equation is
essentially a statement of the conservation of radiative energy. So an important radiant
property is the radiance. The radiance is the radiant power per unit
area in the direction of propagation. And the SI units are watts
per meter squared. Spectral irradiance is the irradiance
per unit frequency or wavelength. And so the units would be watt second per
meter squared in the case of frequency, or watts per nanometer per meter
squared in the case of wavelength. Nanometers is not really an SI units,
meters would be the units. But typically, wavelengths and
ultraviolet and visible region are characterized with
nanometers, the unit nanometers. In the infrared region,
one might use microns, for example. So the rate here is
the radiative transfer equation. One assumption that's been
made in this particular form is that the spectral irradiance
is E sub Nu is broader, spectrally, than the absorption
lines in question. And we'll get to what
an absorption line is in a minute. But this is the equation. So it's the rate of change of the spectral
irradiance, E sub Nu with distance. And the first term -hNu(B12N1)
takes into account absorption. The second term which turns out to be
plus because of a minus times a minus, plus hNuB21N2 takes into
account induced emission. Now both those terms are multiplied by a
phi of Nu and by the spectral irradiance. And we know from the Einstein
picture that the rate of absorption, or induced emission,
is proportional to the rate at which photons interact with the atom or
molecule. So phi nu is something called
the line shape factor. We'll get to that in the next slide. Then the third term in the equation
hNu A21 over 4pi times N2 times the line shape factor is
the spontaneous emission term. And spontaneous emission in a collisionly dominated media like a gas that, say, a combustion gas,
occurs in all directions. And so this 1 over 4pi accounts for the fact. And so
this term only considers the photons that are emitted in the direction that we're making the calculation. One can define absorption
emission coefficients. Alpha sub Nu is
the absorption coefficient. And that basically is the coefficient
of E sub Nu the spectral irradiance. And then epsilon sub Nu is
the emission coefficient. And that is the entire
spontaneous emission term. So with those two definitions, you can write the radiative transfer
equation in this very simple form. That the E sub Nu dz is
equal to epsilon sub Nu- alpha sub Nu times E sub
Nu the spectral irradiance. So what's with this
line broadening factor? Since the energy difference between
two quantum states is a precise value, you might expect that only radiation
of frequency, Nu equals E/h where E is the energy difference between the two
quantum states would be absorbed or emitted. Turns out this is not the case. And it is mainly because of two affects,
the so-called Doppler effect and collisions that the actual absorption
emission line appears to be broader. The Doppler effect is the perceived
change in the frequency of an emitting object due to changes in the relative
velocity with respect to an observer. This is best known from the change
in tone of a car horn or train whistle as a vehicle passes by. [NOISE] The Doppler effect
line width is derived from Maxwellian's velocity distribution. The Doppler effect, of course, is taking
place because as the train horn blows. If you were sitting in the engine,
you would not hear a frequency change. But the train's coming towards you, so
it appears to be at a higher frequency. And then as the train goes past,
it appears to be at a lower frequency. As the sound that I just
demonstrated shows. So it turns out that that line shape
factor, due to the Doppler effect, can be written as shown in
this equation on this slide. Now, collisional broadening comes
about for a different reason. Light or
radiation is electromagnetic radiation. It's described by Maxwell's equations. And a simple, coherent,
single frequency light wave can be represented as a continuous sine or
cosine wave. What a detector sees when
the light hits the detector is actually the Fourier transform
of the time-dependent wave. So for a simple wave,
that's just a pure sine or cosine, the Fourier transform is a delta function. However, light is is not emitted by atoms
and molecules without interruptions. Collisions will certainly
disrupt absorption and emission. And emission will only last
the lifetime at most of the inverse of the Einstein A-coefficient. And so it's actually a finite chunk
of a sine or of a cosine wave. And the Fourier transform of
that is not a delta function. It has some width. And one can show that
the line shape function is of the form shown in
the bottom of the equation. This functional form is
called the Lawrencium. So these two effects are shown
together in this plot. The Doppler one which tends to be
narrower and higher if it's normalized. These are normalized plots. And the collisional or
Lawrencium shape function, which is shorter and fatter in frequency. And the convolution of the two which
is what you actually get in practice, called the Voigt profile,
is shorter and fatter yet. So what do we do with this? Well, we can do spectroscopy. Absorption spectroscopy is
where we stick a light in and see how it is diminished by absorption. Typically, the excited state
concentrations are less than the ground state concentrations. So that the radiative transfer
equation can be very and we ignore spontaneous emission. If we ignore spontaneous emission, then the radiative transfer equation
has a very simple solution. It's just a first order ordinary
differential equation de to z. dz is equal to minus alpha times E. And the integral of that
is that E of z is equal to E of 0 times the exponent of or
E to the minus alpha z. And so, basically,
what one does is measure alpha. And of course, alpha contains
the ground state population. And that's called Beer's Law. Emission spectroscopy, on the other hand, just observes radiation from the atoms and
molecules in the media. And so the radiative equation there
is the solution to the radiative equation is just that E sub Nu
of z is equal to, well, for a uniform media,
would be the emission coefficient times z. So one can do a great
deal of spectroscopy, learn a great number of things,
using using this concept. So this is an example of
an absorption spectrum. This is from the website that's
referenced at the bottom of the slide. It's the infrared absorption
spectrum of the vanillin molecule, which is an important molecule
in biomass processing. And it's in the 2.5 to 20 micron range which is called mid ir. And you can see there are many,
many absorption features. And you can learn a number of
things from a spectrum like this. You can get specific data about
the structure of the molecule. You can also get the temperature. It's a little hard on really
big molecules like this. But on smaller molecules by observing the
spectrum you can ratio, use the Boltzmann ratio to ratio line populations and
recover a temperature and so on. So there are a wide variety
of spectroscopic techniques that are quite useful in
a variety of situations.